Integration application question?

Lets represent Sales by S
Then rate of sales change = dS/dt = f(t) = 10t/(2t^2+1)^2 ---- (1)

From this the goal would be to obtain S = S(t)
This can be done using numerical approximation or by finding exact solution using substitution.

First exact solution.
Substitute Z = 2t^2 +1 ..... (2)
which mean dZ = 4t dt .... (3)
From equation (1), (2) and (3) we obtain
dS=(5/2)*(Z^-2)dZ ... .(4)
Direct integraton results in
S = (-5/2Z)+C ... (5) where C is integration constant
From equation (2) and (5) we have
S = (-5/2*(2t^2 +1))+C .... (6)

But at t =0, there are no sales and hence S =0 (initial condition)
Substituting initial condition in (6), we have the equation for sales as

S = 2.5 (1-1/(2t^2 +1)) ..... (7)

Calculating sales at end of every year and hence sales per year, yeilds following table

t (years) Total Sales *1E5 Sales in Year *1E5
1 1.67 1.67
2 2.22 0.56
3 2.37 0.15
4 2.42 0.06
5 2.45 0.03
6 2.47 0.01

This shows that
(a) Around 56000 models are sold in year 2
and
(b) The model will not be sold in year 5 as the sales in year 4 are below 6000 (and hence < 10000)


Second Numerical Approach:
I simply made an excel function for equation (7) using first order backward difference scheme using a step size of 0.001 years. This would also yield the same result as above. See excel VBA function below..

Function TotalSalesForYears(t As Double) As Double
Dim Step, time, Slope, TotalSales As Double
Step = 0.001
time = 0
TotalSales = 0
Do
Slope = 10 * time / ((2 * time * time + 1) ^ 2)
TotalSales = TotalSales + Step * Slope
time = time + Step
Loop While (time <= t)
TotalSalesForYears = TotalSales
End Function

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